There are two main ways to structure a series: each book is essentially a stand-alone with a continuing story as part of the plot (Harry Potter), or each book is a critical part of the whole and they are difficult to read out of sequence or without the other books (Lord of the Rings). Either is fine; they just accomplish different things.
Stand-alone books have the benefit of being easy to draw in new readers out of sequence, as Shan observes. They don't have an overall story to tell as much as they are a journey being taken with the same characters. Harry's story does have a "proper resolution" in the long view, but each book resolves its individual crises, particularly the first three.
A series of books is one story being told in many parts. Each book does need some kind of "beginning, middle, and end," but by no means does each book require closure. Fellowship of the Ring ends with the Fellowship broken: one of the Nine Walkers dead and one presumed dead, two kidnapped, and two vanished off to Mordor. But the story has progressed from "One Ring's discovery" to "Frodo accepts the quest" to "Frodo and Sam leave for Mordor." The "proper resolution" doesn't occur until the multiple ends ;) of Return of the King.
I agree with Shan that if I'm in a bookstore and I see "Book 1 of 3" I may not pick it up unless Books 2 and 3 are next to it on the shelf. However, don't let that stop you from telling your story in the way that your story needs to be told. Write the best story you can, structure it however makes it the most appealing, and let the marketing folks at the publisher worry about selling it in pieces.
David Eddings originally planned his Belgariad series as a trilogy, and the publisher broke it into five books. George R.R. Martin planned the Song of Ice and Fire story for four novels, which grew into six and then split into seven. Yep, pain in the butt for us readers now, but when it's all over, they will all be together on one shelf for future generations.